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Deterministic Realization of Collective Measurements Via Photonic Quantum Walks

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Deterministic Realization of Collective Measurements Via Photonic Quantum Walks ARTICLE DOI: 10.1038/s41467-018-03849-x OPEN Deterministic realization of collective measurements via photonic quantum walks Zhibo Hou1,2, Jun-Feng Tang1,2, Jiangwei Shang3,4, Huangjun Zhu5,6,7,8,9, Jian Li10,11, Yuan Yuan1,2, Kang-Da Wu1,2, Guo-Yong Xiang1,2, Chuan-Feng Li1,2 & Guang-Can Guo1,2 Collective measurements on identically prepared quantum systems can extract more infor- mation than local measurements, thereby enhancing information-processing efficiency. 1234567890():,; Although this nonclassical phenomenon has been known for two decades, it has remained a challenging task to demonstrate the advantage of collective measurements in experiments. Here, we introduce a general recipe for performing deterministic collective measurements on two identically prepared qubits based on quantum walks. Using photonic quantum walks, we realize experimentally an optimized collective measurement with fidelity 0.9946 without post selection. As an application, we achieve the highest tomographic efficiency in qubit state tomography to date. Our work offers an effective recipe for beating the precision limit of local measurements in quantum state tomography and metrology. In addition, our study opens an avenue for harvesting the power of collective measurements in quantum information- processing and for exploring the intriguing physics behind this power. 1 Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, P. R. China. 2 Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, P. R. China. 3 Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Siegen 57068, Germany. 4 Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, Beijing 100081, China. 5 Institute for Theoretical Physics, University of Cologne, Cologne 50937, Germany. 6 Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China. 7 Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China. 8 State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China. 9 Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China. 10 Institute of Signal Processing Transmission, Nanjing University of Posts and Telecommunications, Nanjing 210003, China. 11 Key Lab of Broadband Wireless Communication and Sensor Network Technology, Nanjing University of Posts and Telecommunications, Ministry of Education, Nanjing 210003, China. Correspondence and requests for materials should be addressed to H.Z. (email: [email protected]) or to G.-Y.X. (email: [email protected]) NATURE COMMUNICATIONS | (2018) 9:1414 | DOI: 10.1038/s41467-018-03849-x | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03849-x uantum measurements are the key for extracting infor- merit without using adaptive measurements. Such high efficiency Qmation from quantum systems and for connecting the demonstrates the main advantage of collective measurements quantum world with the classical world. Understanding over separable measurements. Here, we encode the two qubits in – the power and limitation of measurements is of paramount the two degrees of freedom of a single photon24 27, but our importance not only to foundational studies, but also to many method for performing collective measurements can be general- applications, such as quantum tomography, metrology, and ized to two-photon two-qubit states by combining the technique – communication1 8. An intriguing phenomenon predicted by of quantum joining28 or teleportation29. quantum theory is that collective measurements on identically prepared quantum systems may extract more information than Results local measurements on individual systems, thereby leading to – Optimized collective measurements. In quantum theory, a higher tomographic efficiency and precision9 14. The significance measurement is usually represented by a positive-operator-valued of collective measurements for multiparameter quantum measure (POVM), which is composed of a set of positive metrology was also recognized recently15,16. This nonclassical operators that sum up to the identity. In traditional quantum phenomenon is owing to entanglement in the quantum mea- information-processing, measurements are performed on indivi- surements instead of quantum states. It is closely tied to the dual quantum systems one by one, which often cannot extract phenomenon of “nonlocality without entanglement”17. In addi- information efficiently. Fortunately, quantum theory allows us to tion, collective measurements are very useful in numerous other perform collective measurements on identically prepared quan- tasks, such as distilling entanglement18, enhancing nonlocal tum systems in a way that has no classical analog, as illustrated in correlations19, and detecting quantum change point20. However, Fig. 1. demonstrating the advantage of collective measurements in In the case of a qubit, a special two-copy collective POVM was experiments has remained a daunting task. This is because most highlighted in refs.11,13,14, which consists of five POVM elements, optimized protocols entail generalized entangling measurements ED 2 on many identically prepared quantum systems, which are very 3 ð Þ Ej ¼ ψ ψ ; E5 ¼ jiΨÀ hjΨÀ ; 1 difficult to realize deterministically. 4 j j Here we introduce a general method for performing determi- jiΨ ¼ p 1ffiffi ðÞEjiÀ ji nistic collective measurements on two identically prepared qubits where À 01 10 is the singlet, which is maximally ψ2 = based on quantum walks, which extends the method for per- entangled, and j for j 1, 2, 3, 4 are qubit states that form a – forming generalized measurements on a single qubit only21 23. symmetric DE informationally complete POVM (SIC-POVM), that 2 By devising photonic quantum walks, we realize experimentally is, ψ ψ = (2δ + 1)/3E 30,31. Geometrically, the Bloch vec- j k jk a highly efficient collective measurement highlighted in ψ 11,13,14 fi tors of the four states j form a regular tetrahedron inside the refs. As an application, we realize, for the rst time, qubit Bloch sphere. For concreteness, here we choose state tomography with deterministic collective measurements. ÀÁpffiffiffi The protocol we implemented is significantly more efficient than ψ ¼ ji0 ; ψ ¼ p1ffiffi ji0 þ 2ji1 ; 1 ÀÁ2 3 pffiffiffi local measurements commonly employed in most experiments. 2π ψ ¼ p1ffiffi jiþ 3 i ji; ð Þ Moreover, it can achieve near-optimal performance over all two- 3 0 e 2 1 2 3 fi ÀÁπ pffiffiffi copy collective measurements with respect to various gures of p1ffiffi À2 i ψ ¼ jiþ0 e 3 2ji1 : 4 3 a N b c N/2 d C (–2,2) C (–2,4) –2 E4 –1 C (–1,1) C (–1,3) C (–1,5) C (0,2) C (0,4) 0 E3 C (1,3) 1 E5 C (2,2) 2 E2 3 E1 t =1 t =2 t =3 t =4 t =5 Fig. 1 Individual and collective measurements. a Repeated individual measurements. b Single N-copy collective measurement. c Repeated two-copy collective measurements. d Realization of the collective SIC-POVM defined in Eqs. (1) and (2) using five-step quantum walks. The coin qubit and the walker in positions 1 and −1 are taken as the two-qubit system of interest, whereas the other positions of the walker act as an ancilla. Site-dependent coin operators C(x, t) are specified in the Methods section. Five detectors E1 to E5 correspond to the five outcomes of the collective SIC-POVM 2 NATURE COMMUNICATIONS | (2018) 9:1414 | DOI: 10.1038/s41467-018-03849-x | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03849-x ARTICLE The POVM defined by Eqs. (1) and (2) is referred to as the operators C(x, t) followed by measuring the walker position after collective SIC-POVM henceforth. If this POVM is performed on certain steps. However, little is known in the literature on rea- ⊗ the two-copy state ρ 2, then the probability of obtaining outcome lizing POVMs on higher-dimensional systems. Here, we propose = ρ⊗2 j is given by pj tr( Ej). a general method for extending the capabilities of quantum walks. The collective SIC-POVM is distinguished because it is optimal For concreteness, we illustrate our approach with the collective in extracting information from a pair of identical qubits9,11.Itis SIC-POVM. universally Fisher symmetric in the sense of providing uniform To realize the collective SIC-POVM using quantum walks, the and maximal Fisher information on all parameters that coin qubit and the walker in positions 1 and −1 are taken as the characterize the quantum states of interest13,14,32. Moreover, it two-qubit system of interest, whereas the other positions of the is unique such POVM with no more than five outcomes. walker act as an ancilla. With this choice, the collective SIC- Consequently, the collective SIC-POVM is significantly more POVM can be realized with five-step quantum walks, as efficient than any local measurement in many quantum illustrated in Fig. 1d and discussed in more details in information-processing tasks, including tomography and metrol- Supplementary Note 1. Here, the nontrivial coin operators C(x, fi fi fi ogy. Moreover, its high tomographic ef ciency is achieved t) are speci ed in the Methods section. The ve detectors E1 to E5 without using adaptive measurements, which is impossible for marked in the figure correspond to the five POVM elements local measurements. As far as two-copy collective measurements specified in Eqs. (1) and (2). Moreover, this proposal can be are concerned, surprisingly, more entangled measurements, such implemented using photonic quantum walks, as illustrated in as the Bell measurements, cannot lead to higher efficiency. Fig. 2 (see also Supplementary Fig. 1). Although multi-copy (say three-copy) collective measurements can further improve the efficiency, the improvement is not so Experimental setup. The experimental setup for realizing the significant13,14. collective SIC-POVM and its application in quantum state tomography is presented in Fig. 2. The setup is composed of two Realization of the collective SIC-POVM via quantum walks.
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